On Gao’s model of wavefunction collapse Lajos Di´ osi Wigner Centre, Budapest 13 Oct 2018, Taiyuan Acknowledgements go to: Shanxi University National Research Development and Innovation Office of Hungary, grant K12435 EU COST Action CA15220 ‘Quantum Technologies in Space’
Based on his RDM interpretation of the wavefunction, Gao has constructed a simple discrete-time energy-conserving model of spontaneous (i.e.: objective) wavefunction collapse. I recast, equivalently, the stochastic equations of this model and I discuss it in the context of alternative collapse models like, e.g., Pearle’s gambler’s ruin process, previous energy-driven collapse models and the gravity-related model of Penrose and myself. Background concepts Collapse as Gambler’s Ruin Gao’s Collapse Model Diffusive Limit Two-State Eample Diffusive Limit for Full Denisty Matrix Decoherence time, collapse time Physics of Collapse Summary
Background concepts What I read from Gao: ◮ Unitary evolution of Ψ and its Born’s probability densities are underlied by ergodic random discontinuous motion (RDM) of the particles. ◮ Also non-unitary collapse dynamics of Ψ is discontinuous. ◮ The chosen collapse model is discrete in time. I take these for granted, as a possible alternative to other spontaneous collapse theories .
Collapse as Gambler’s Ruin Collapse + Born’s rule: N � | Ψ � = c k | k � k = 1 | c k | 2 ( final ) = δ kn , with probability | c n | 2 ( initial ) . Pearle’s hint of a stochastic model: N gamblers with initial moneys p 1 = | c 1 | 2 , p 2 = | c 2 | 2 , etc . play a fair game which ends when all gamblers go to ruin except for the winner who takes everything. A possible fair game (there are many more): 1. They put λ p 1 , λ p 2 , etc . into the bank ( λ ≤ 1 ) 2. Bank money λ is given to player n with probability p n p k → ( 1 − λ ) p k + δ kn λ 3. Go to 1. until one player wins everything. Then p k ( final ) = δ kn , with probability p n ( initial ) . That’s collapse + Born rule.
Gao’s Collapse Model N � | Ψ( 0 ) � = c k ( 0 ) | E k � , ̺ ( 0 ) = | Ψ( 0 ) �� Ψ( 0 ) | ˆ k = 1 � � ̺ ( t ) = ˆ p k ( t ) | E k �� E k | + ̺ jk ( t ) | E j �� E k | k j � = k Discrete stochastic dynamics ˆ ̺ ( t ) → ˆ ̺ ( t + t Pl ) : ̺ ( t + t Pl ) = ( 1 − λ )ˆ ˆ ̺ ( t ) + λ | E n �� E n | , with probability p n ( t ) = � E n | ˆ ̺ ( t ) | E n � . ⇒ p k ( t + t Pl ) = ( 1 − λ ) p k ( t ) + λδ kn , ̺ jk ( t + t Pl ) = ( 1 − λ ) ̺ jk ( t ) � �� � � �� � like in gambler’s ruin ∗ damping (decoherence) p k ( ∞ ) = δ kn with probability p n ( 0 ) ̺ jk ( ∞ ) = 0 ( j � = k )
Diffusive Limit During ∆ t = t Pl : discrete change ∆ p k ≡ ∆ p k | n = − λ ( p k − δ kn ) with prob. p n . Let’s calculate 1st & 2nd moments of ∆ p k | n : � E ∆ p k | n = n p n ∆ p k | n = 0 E ∆ p j | n ∆ p k | n = � n p n ∆ p j | n ∆ p k | n = λ 2 ( p j δ jk − p j p k ) On scales t >> t Pl : inhomogeneous diffusion , with diff. matrix t − 1 Pl λ 2 ( p j δ jk − p j p k ) . 1. Ito formalism, with { ξ k } white noises: � dp k = λ ( p k d ξ n − d ξ k ) n E d ξ j d ξ k = t − 1 E d ξ k = 0 , Pl p j δ jk dt 2. Fokker-Planck formalism, for density ̺ ( p 1 , p 2 , . . . ; t ) : ̺ ( p 1 , p 2 , . . . ; 0 ) = � N k = 1 δ ( p k − p k ( 0 )) ∂ t = λ 2 ∂ 2 ∂̺ � ( p k δ jk − p j p k ) ̺ t Pl ∂ p j ∂ p k jk
Two-State Example Single variable q = p 1 − p 2 , q ∈ [ − 1 , + 1 ] : p 1 = ( 1 + q ) / 2 , p 2 = ( 1 − q ) / 2 Take initial density ̺ ( q , 0 ) = δ ( q − p 1 ( 0 ) + p 2 ( 0 )) . Fokker-Planck eq. reduces to = λ 2 ∂ 2 ∂̺ ( q , t ) ∂ q 2 ( 1 − q 2 ) ̺ ( q , t ) . ∂ t t Pl ⇒ ̺ ( q , ∞ ) = p 1 ( 0 ) δ ( q − 1 ) + p 2 ( 0 ) δ ( q + 1 ) That’s collapse + Born rule. Pearle-Gisin version: = λ 2 ∂ 2 ∂̺ ( q , t ) ∂ q 2 ( 1 − q 2 ) 2 ̺ ( q , t ) ∂ t 2 t Pl
Diffusive Limit for Full Density Matrix
Decoherence time, collapse time � � 1 0 0 0 � � � � p 1 ̺ 12 p 1 0 decoherence collapse − − − − − − → − − − − → ̺ 21 p 2 0 p 2 τ D τ C � � 0 0 0 1 Decoherence is mandatory for collapse. Decoherence process is falsifiable, collapse process is not. (D) in known spontaneous collapse theories Decoherence can be much faster than collapse ( τ D ≪ τ C ) : if λ ≪ 1 � � τ D = 1 τ C = 1 λ t Pl , λ 2 t Pl ( λ < 1 ) . With Gao’s choice λ = ∆ E / E Pl (valid for ∆ E ≤ E Pl ): τ C = � E Pl (∆ E ) 2 , ∆ E = energy spread . Coincides with collapse time in old energy-driven models (Percival, Hughston, Milburn).
Physics of Collapse Relevance of τ C = � E Pl (∆ E ) 2 τ C = 10 13 s (irrelevant) ∆ E = 1eV (atomic superposition) τ C = 10 − 5 s (irrelevant) ∆ E = 1GeV (high energy superposition) τ C = 10 − 25 s (killing) ∆ E = 1J (macroscopic superposition) Gao: ∆ E is not defined as the uncertainty of the total energy of all sub-systems. [...] each sub-system has its own energy uncertainty that drives its collapse ... provided system splits into non-interacting subsysems. If they interact, collapse’s energy-conservation will be gone. CSL and D-Penrose theories prescribe collapses to local mass densities, hence they can not preserve energy.
Summary I showed that Gao’s model: ◮ is a simple gambler’s ruin process ◮ has a diffusive limit, similar to (but different from) the Pearle-Gisin collapse ◮ yields collapse time formally equal to old energy driven/conserving models ◮ and improves them by the statement of subsystem-wise collapses In the future, it ◮ can be further discussed in the zoo of spontaneous collapse theories ◮ might lead to similar kind of testable predictions ◮ will face similar difficulties
Recommend
More recommend
